A mathematical term describing a particular type of growth or decay. Other types of growth patterns are possible and include linear, polynomial, or even super-exponential.
Just because something is growing rapidly does not mean that it is growing exponentially. To be growing exponentially, something has to increase by the same percentage in any set of intervals of equal duration. If the world population were to grow exponentially by 1% a year, every year the population would be 1% larger than the previous year. The years become the set of intervals of equal duration (1 year).
No physical entity can grow exponentially forever. There are many exercises for students that can illustrate this principle. For instance, if the average person contains 50 kg of water, the current population of the earth is 7 billion and growing at a rate of 1 % (a little low), and there are 1.4×1021 kg of water on the earth, how long before all the water is incorporated into people? (Answer at the end of this article)
Just because something is decaying does not mean that is it is decaying exponentially. When something decays exponentially, it is reduced by the same percentage in any set of intervals of equal duration. If something is decaying exponentially, it gets smaller and smaller and approaches zero, but mathematically it never actually gets to zero.
If you are not sure about the growth pattern, or the decay pattern, why not simply say “growing rapidly” or “decaying quickly.” The statement that X is growing exponentially, is a very specific claim.
Example: Wind Energy
The US Energy Information Agency’s Dec 2016 Short term energy outlook provides the following data for US renewable energy supply for wind in Quadrillion Btu from 2008 to 2015.
| year | Quadrillion Btu | Increase | % Increase |
| 2008 | 0.546 | ||
| 2009 | 0.721 | 0.176 | 32% |
| 2010 | 0.923 | 0.202 | 28% |
| 2011 | 1.167 | 0.244 | 26% |
| 2012 | 1.339 | 0.172 | 15% |
| 2013 | 1.600 | 0.261 | 19% |
| 2014 | 1.726 | 0.126 | 8% |
| 2015 | 1.812 | 0.086 | 5% |
The column labeled increase is just the value for a given year minus the previous year’s value. so for 2011 the increase was equal to 1.167 (2011) minus 0.923 (2010) or 0.244 (the difference). (I have done the calculation in Excel using the values from the Energy Information Agency’s website, which are not rounded.) If this was a perfect linear increase, all the values in this column would be the same. The values bounce around 0.2 quite a bit, except for the 2015 value. The column labeled % increase is the increase divided by the previous years data. So for 2011 the percent increase is 0.244 divided by 0.923 (the value for 2010). This yields 0.264 which is not a percentage. To express this fraction in percent multiply it by 100, yielding 26% (rounded). If this were an example of purely exponential growth all the values in this column would be the same. Looking at the data, the % increase is steadily decreasing! This suggests that this is not an example of exponential growth.
In school I was taught, and I taught my students, to always plot the data! So here is a plot of the data.
I have also asked Excel to “fit” a linear trend line to the data. This procedure uses a mathematical algorithm to put the “best” straight line though the data. Here best means that the “sum of the errors” is as small as possible. To find the sum of the errors, the vertical distances between the data points and the line are squared, then all these squares are added together. The equation that Excel has placed on the graph is the mathematical equation of the best fit straight line. The value of R2 gives us a relative score for the “goodness of fit.” This value is useful to compare various types of fits. A perfect fit would give and R2 of 1. A value of almost 0.99 is pretty good.
If we look carefully at the graph, the last three points are all in a line, but that line is less steep than the fit. This actually proves very little, the next point (2016) could easily come above the best fit value.
If we try the same experiment with an exponential fit (exponential growth) Excel returns the following graph.
Visually you can see that on average the data points fall farther from the curve. The displacement of the data from the fit form a pattern, only the middle points lie above the curve. The fit also has a lower R2 value. Taken together this suggests that at least for this data, the linear fit is probably better than the exponential.
The type of growth assumed makes a big difference when the fit is used to project the future. In the graph below both the linear fit and the exponential fit are extended to 2050, looking at the implications for the future suggest by each fit.
The US uses about 100 Quad Btu’s a year, so I have limited the vertical axis to that amount. If the linear trend continues, in 2050 wind would amount to just under 10% of the current US energy budget. If the exponential growth curve would be followed, 100% of the current US energy budget would be supplied by wind power around 2035. The two fits predict vastly different outcomes .
What future depends on the choices we make. For wind to produce 100 Quads of energy requires about 4,000,000 MW of installed wind generation capability(1). According to the wind industry installing each MW of wind capability costs between 1.3 and 2.2 million dollars. (2) The smallest cost to install all of this wind power is about $1,300,000 ($/per MW) times 4,000,000 MW or 5.2 trillion dollars. This does not include the cost of any added transmission lines, transformers, maintenance, or energy storage requirements to average out the varying wind.
The cost would be invested over time, but since the growth in wind is exponential, so is the cost! The plot below estimates the required investment in each year as the assumed exponential growth in wind power unfolds (using the $1,300,000 per MW number).
As the graph illustrates, as exponential growth continues, so does the cost of maintaining that growth. If the growth rate in wind energy is faster than the growth rate in Gross National Product, larger and larger fractions of the US economy are required to maintain that growth. If there are no other physical limitations, the economic ones will ultimately limit the growth.
Which model gives better predictions? One of the things scientists look for in fitting data is to find some explanation that suggests one fit over the other. Since this data is the result of human decision-making, such an explanation would not be uncovered by only looking at this data. What happens in the future is dependent on the choices that humans make. These fits only help us analyze the past and play “what – if” games with the future. The best fit to the past data, seems to be a linear one.
Oh, the answer to the puzzle? 2,211 years from now. This is surprisingly short, illustrating the problem with exponential growth models. This example illustrates that there are always physical limits to the exponential growth of physical systems.
(1) From Wind Energy as a Significant Source of Electricity R. Gerald Nix NREL/TP-441-8162 downloaded from http://www.nrel.gov/docs/legosti/old/8162.pdf on 12/16/2016
(2) http://www.windustry.org/how_much_do_wind_turbines_cost accessed on 12/16/2016
Last update: Dec 16, 2016